Is 65536 the only power of 2 without digits 1,2,4,8?

//This program checks values n up to 10^10 for which the last (up to) 51 digits of 16^n are all 0,3,5,6,7,9.
//This program was written to attempt to find a power of 2 other than 65536 which does not contain any digit 1,2,4,8.
//This program only finds two candidates (n=7525657339 and 9881236075) but checking their first few digits via the logarithm method shows that they don't work.
//Based on http://tasvideos.org/forum/viewtopic.php?p=437945#437945

#include <stdio.h>
#include <stdint.h>
#include <cmath>

int main()
{
    uint64_t val1=1ULL;
    uint64_t val2=0ULL;
    uint64_t val3=0ULL;
    uint64_t carry1,carry2;

    for(uint64_t i=1;i<=10000000000ULL;i++)
    {
        val1*=16ULL;
        carry1=val1/100000000000000000ULL;
        val1%=100000000000000000ULL;
        val2=val2*16ULL+carry1;
        carry2=val2/100000000000000000ULL;
        val2%=100000000000000000ULL;
        val3=val3*16ULL+carry2;
        val3%=100000000000000000ULL;

        //check last 17 digits of val1 are all 0,3,5,6,7,9

        uint64_t c=val1;
        //ones digit is always 6
        c/=10ULL;

        int flag=1;
        for(int j=1;j<=16;j++)
        {
            int digit=(int)(c%10ULL);
            if(digit==1 || digit==2 || digit==4 || digit==8)
            {
                flag=0;
                break;
            }
            c/=10ULL;
        }
        if(flag==1)
        {
            //check last 17 digits of val2 are all 0,3,5,6,7,9

            c=val2;
            for(int j=1;j<=17;j++)
            {
                int digit=(int)(c%10ULL);
                if(digit==1 || digit==2 || digit==4 || digit==8)
                {
                    flag=0;
                    break;
                }
                c/=10ULL;
            }
            if(flag==1)
            {
                //check last 17 digits of val3 are all 0,3,5,6,7,9


                c=val3;
                for(int j=1;j<=17;j++)
                {
                    int digit=(int)(c%10ULL);
                    if(digit==1 || digit==2 || digit==4 || digit==8)
                    {
                        flag=0;
                        break;
                    }
                    c/=10ULL;
                }
                if(flag==1)
                    printf("%llu %llu %017llu %017llu\n",i,val3,val2,val1);

            }




        }

        if(i%100000000ULL==0) printf(".");

    }

    printf("Done!");
    getchar();
    return 0;

}